Answer
$\dfrac{39-13\sqrt{11}}{-2}$
Work Step by Step
RECALL:
(i) For any nonnegative real numbers $a$ and $b$, $(a - \sqrt{b})(a+\sqrt{b}) = a^2-b$.
(ii) For any nonnegative real numbers a and b, $\sqrt{a}\cdot\sqrt{b} = \sqrt{ab}$.
Rationalize the denominator by multiplying $3-\sqrt{11}$ to the numerator and the denominator. Then, use rule (i) above to obtain
$\dfrac{13}{3+\sqrt{11}}\cdot \dfrac{3-\sqrt{11}}{3-\sqrt{11}}
\\=\dfrac{13(3-\sqrt{11})}{3^2-11}
\\=\dfrac{13(3) - 13\sqrt{11}}{9-11}
\\=\dfrac{39-13\sqrt{11}}{-2}.$
